Subadditive Theorem Research Articles

Subadditive theorems in time-dependent environments

Subadditive theorems in time-dependent environments

On Subadditivity of Okounkov Bodies for Algebraic Fiber Spaces

Abstract The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.

On subadditive theorems for group actions and homogenization

We prove subadditive theorems à la Akcoglu-Krengel on measure spaces with acting groups. Applications to homogenization of nonconvex integrals in Cheeger-Sobolev spaces are also developed.

Positivity of the time constant in a continuous model of first passage percolation

For a given dimension d $\ge$ 2 and a finite measure $\nu$ on (0, +$\infty$), we consider $\xi$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $\nu$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $\Sigma$ = $\cup$ (c,r)$\in$$\xi$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $\Sigma$ and at infinite speed inside $\Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $\mu$ x when x goes to infinity, where $\mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $\mu$ as a function of the measure $\nu$ associated with the underlying Boolean model.

Subadditivity of chance measure

Chance theory is a mathematical methodology for dealing with indeterminacy phenomena involving uncertainty and randomness. In this paper, some properties of chance space are investigated. Based on this, the subadditivity theorem, null-additivity theorem, and asymptotic theorem of chance measure are proved.

The volume of an isolated singularity

We introduce a notion of volume of a normal isolated singularity that generalizes Wahl’s characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of noninvertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a noninvertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov’s b-divisors. We define the notions of nef Weil b-divisors and of nef envelopes of b-divisors. We relate the latter to the pullback of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil b-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case.

Credibility measure of fuzzy sets and applications

In order to measure a fuzzy event, credibility measure is proposed as a non-additive set function. In this paper, this concept is extended to fuzzy sets. First, a mean measure is defined by Lebesgue integral and some properties are investigated such as the monotonicity theorem, self-duality theorem and subadditivity theorem. Furthermore, by using Sugeno integral, an equilibrium measure is proposed and studied. Finally, these concepts are applied to optimisation problems, and the mean measure maximisation model and equilibrium measure maximisation model are proposed.

When does the subadditivity theorem for multiplier ideals hold?

Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.